Systems of Linear Equations

Introduced in Math 115

[! Definition 2.1.2]

A linear equation in n variables is an equation of the form

$a_1{x}_1+a_2{x}_2+\dots a_n{x}_n=b$ Where $x_1,\dots ,x_n\in \mathbb{R}$ are the variables or unknowns, $a_1,\dots ,a_n\in \mathbb{R}$ are coefficients and b $\in \mathbb{R}$ is the constant term. A system of linear equations (also called a linear system of equations) is a collection of finitely many linear equations

A linear system of equations is considered consistent if it has at least one solution. Otherwise, we call the linear system inconsistent

[! Definition 2.2.5]

The first nonzero entry in each row of a matrix is called a leading entry (or a pivot). A matrix is in Row Echelon Form (REF) if

a) All rows whose entries are all zero appear below all rows that contain nonzero entries

b) Each leading entry is to the right of the leading entries above it A matrix is in Reduced Row Echelon Form (RREF) if it is in REF and

c) Each leading entry is a 1, called a leading one

d) Each leading entry is the only nonzero entry in its column